gpt pro 5.2 attempt on Math-6171

1) FORMAL RESTATEMENT

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function that is not a monomial (i.e., not of the form $c z^m$ with $c\ne 0$, $m\in\mathbb{Z}_{\ge 0}$). For each $r>0$, define \[ M(r):=\max_{|z|=r} |f(z)| \] and let \[ \nu(r):=\#\{z\in\mathbb{C}: |z|=r\text{ and }|f(z)|=M(r)\}. \] (As noted in the problem statement, $\nu(r)$ is finite for non-monomial $f$; we reprove this below.)

Question 1: Can $\limsup_{r\to\infty} \nu(r)=\infty$ occur?

Question 2: Can $\liminf_{r\to\infty} \nu(r)=\infty$ occur?

2) QUICK LITERATURE/CONTEXT CHECK

I did not browse the web; I only record what is stated in the problem text:

- Herzog--Piranian $1968$ showed Question 1 has answer "yes". - Question 2 is still open; Glücksam--Pardo-Simón $2024$ give an "approximate" affirmative answer.

3) ATTACK PLAN

Proof track (for Question 2):

- Try to modify the Herzog--Piranian construction to control $\nu(r)$ from below for all sufficiently large radii $r$, not just along a sparse sequence.

Disproof track (for Question 2):

- Try to prove that every non-monomial entire function admits infinitely many radii with $\nu(r)$ bounded (e.g. $\nu(r)\le C$ for infinitely many $r$), using growth/regularity theorems for entire functions.

Below I only establish basic finiteness and a polynomial bound; these do not resolve Question 2.

4) WORK

Lemma 1117.1 (Real-analyticity of the boundary modulus). Let $f$ be entire and fix $r>0$. The function \[ \theta\mapsto f(r e^{i\theta}) \] is complex-analytic as a function of $e^{i\theta}$, and in particular the real-valued function \[ g_r(\theta):=|f(r e^{i\theta})|^2 \] is a $2\pi$-periodic real-analytic function of $\theta$.

Proof. Write the power series $f(z)=\sum_{n=0}^\infty c_n z^n$, which converges uniformly on $|z|\le r$ since $f$ is entire. Then \[ f(r e^{i\theta}) = \sum_{n=0}^\infty c_n r^n e^{i n\theta}, \] with uniform convergence in $\theta$. This is a uniformly convergent Fourier series with only nonnegative frequencies, hence defines a real-analytic function of $\theta$ (indeed it extends to a complex-analytic function of $\theta$ in a horizontal strip). Taking modulus squared preserves real-analyticity, so $g_r$ is real-analytic and $2\pi$-periodic. $\square$

Lemma 1117.2 (Constant modulus on a circle forces a monomial). Let $f$ be entire. Suppose there exists $r>0$ and a constant $C>0$ such that \[ |f(z)|=C\quad\text{for all }|z|=r. \] Then $f$ is a monomial: $f(z)=c z^m$ for some integer $m\ge 0$ and constant $c\ne 0$.

Proof. Define $g(z):=f(r z)/C$. Then $g$ is entire and satisfies $|g(z)|=1$ for all $|z|=1$. Because $g$ is analytic on a neighborhood of the closed unit disk and not identically zero, its zero set inside $|z|\le 1$ is finite (zeros of a nonzero analytic function are isolated, and a compact set contains only finitely many isolated points). Let $a_1,\dots,a_s$ be the zeros of $g$ in $|z|<1$, listed with multiplicity. Define the finite Blaschke product \[ B(z)=\prod_{j=1}^s \frac{z-a_j}{1-\overline{a_j}z}. \] This $B$ is analytic on a neighborhood of $|z|\le 1$, satisfies $|B(z)|=1$ for $|z|=1$, and has exactly the same zeros (with multiplicity) in $|z|<1$ as $g$.

Now $h(z):=g(z)/B(z)$ is analytic and nonvanishing on a neighborhood of $|z|\le 1$, so both $h$ and $1/h$ are analytic on $|z|\le 1$. On $|z|=1$ we have $|h(z)|=|g(z)|/|B(z)|=1$. By the maximum modulus principle applied to $h$ and to $1/h$ on the unit disk, we obtain $|h(z)|\le 1$ and $|h(z)|\ge 1$ for $|z|<1$. Hence $|h(z)|=1$ on the open unit disk. An analytic function with constant modulus on a domain is constant (e.g. by the open mapping theorem), so $h\equiv e^{i\phi}$ for some real $\phi$. Thus, on $|z|\le 1$, \[ g(z)=e^{i\phi} B(z). \] Since both sides are analytic on a neighborhood of $|z|\le 1$, the identity theorem extends this equality to that neighborhood.

Finally, because $g$ is entire, the right-hand side must be entire. A Blaschke factor $\frac{z-a}{1-\overline{a}z}$ with $a\ne 0$ has a pole at $z=1/\overline{a}$, which lies outside the unit disk. Therefore, for $e^{i\phi}B(z)$ to be entire, every $a_j$ must equal $0$. Hence $B(z)=z^s$, and $g(z)=e^{i\phi} z^s$ for all $z$. Undoing the definition of $g$, we get $f(z)=C e^{i\phi} (z/r)^s$, i.e. $f(z)=c z^m$ with $m=s$. $\square$

Lemma 1117.3 (Finiteness of $\nu(r)$ for non-monomial entire functions). If $f$ is entire and not a monomial, then for every $r>0$ the number $\nu(r)$ is finite.

Proof. Fix $r$. By Lemma 1117.1, $g_r(\theta)=|f(r e^{i\theta})|^2$ is a real-analytic periodic function. If $g_r$ were constant, then $|f|$ would be constant on $|z|=r$, forcing $f$ to be a monomial by Lemma 1117.2, contrary to hypothesis. Hence $g_r$ is non-constant. A non-constant real-analytic function on a compact interval has only finitely many critical points: its derivative $g_r'(\theta)$ is real-analytic and not identically zero, so its zeros are isolated, hence finite in $[0,2\pi]$. Every maximum point of $g_r$ is a critical point, so there are finitely many maxima, hence finitely many points $z$ with $|z|=r$ attaining $M(r)$. Therefore $\nu(r)<\infty$. $\square$

Lemma 1117.4 (Polynomial bound). If $f$ is a polynomial of degree $n$ which is not a monomial, then $\nu(r)\le 2n$ for every $r>0$.

Proof. Write $f(z)=\sum_{j=0}^n c_j z^j$. Then $f(r e^{i\theta})=\sum_{j=0}^n c_j r^j e^{ij\theta}$ is a trigonometric polynomial of degree $n$, and $g_r(\theta)=|f(r e^{i\theta})|^2$ is a real trigonometric polynomial with frequencies in $[-n,n]$. Its derivative $g_r'(\theta)$ is a real trigonometric polynomial of degree at most $n$. If $g_r'\equiv 0$, then $g_r$ is constant, forcing $f$ to have constant modulus on $|z|=r$, hence to be a monomial by Lemma 1117.2 (applied to the entire function $f$), contradicting the hypothesis. Therefore $g_r'$ is a nonzero trigonometric polynomial of degree at most $n$, so it has at most $2n$ zeros in $[0,2\pi)$. The maximum points of $g_r$ lie among the zeros of $g_r'$, so there are at most $2n$ such points. Hence $\nu(r)\le 2n$. $\square$

FAST REALITY CHECK (a simple explicit example). Let $f(z)=1+z+z^2$. For any $r>0$ and $|z|=r$, the triangle inequality gives $|f(z)|\le 1+|z|+|z|^2 = 1+r+r^2$, with equality only when $z$ has argument $0$ (since all coefficients are positive real). Thus $M(r)=1+r+r^2$ and $\nu(r)=1$ for all $r>0$. A numerical sampling check at $r\in\{0.5,1,2\}$ (dense grid of angles) agrees with $\nu(r)=1$.

5) VERIFICATION

- Lemma 1117.2: the key steps are (a) finitely many zeros in $|z|\le 1$, (b) constructing the finite Blaschke product with the same interior zeros, and (c) applying maximum modulus to $h$ and $1/h$. Each step was justified. - Lemma 1117.3 depends only on real-analyticity plus the non-constantness ensured by Lemma 1117.2. - Lemma 1117.4 uses the fact that a nonzero trigonometric polynomial of degree $n$ has at most $2n$ zeros on a period; this is standard and follows from writing it as a polynomial in $e^{i\theta}$ of degree at most $n$ and applying the fundamental theorem of algebra.

6) FINAL

UNRESOLVED

(i) Strongest proved partial result: $\nu(r)$ is finite for each $r$ when $f$ is not a monomial, and for polynomials of degree $n$ one has the uniform bound $\nu(r)\le 2n$ for all $r$ (Lemmas 1117.3--1117.4).

(ii) First gap (crisp): Either construct an explicit entire non-monomial $f$ such that $\nu(r)\to\infty$ along all sufficiently large radii (i.e. $\liminf_{r\to\infty}\nu(r)=\infty$), or prove that every such $f$ has infinitely many radii $r$ with $\nu(r)$ bounded.

(iii) Top 3 next moves: 1. Study (and attempt to re-derive) the Herzog--Piranian construction that makes $\limsup\nu(r)=\infty$, and isolate exactly where the radii with large $\nu(r)$ occur; try to densify that set of radii. 2. Attempt a "stability" lemma: if $\nu(r)$ is large at some radius, can one force it to remain large on an interval of radii $[r,r+\Delta]$? Proving such persistence would move from limsup to liminf. 3. Numerically explore candidate entire functions with many comparable dominant terms on $|z|=r$ (e.g. lacunary power series with carefully tuned coefficients) and track $\nu(r)$ as $r$ varies, looking for patterns that might suggest a constructive approach to liminf.

(iv) Minimal counterexample structure (if liminf cannot be infinite): one would expect a theorem producing infinitely many radii where the maximum modulus is attained essentially by a single "dominant direction" (few maxima), perhaps tied to regularity of the maximum modulus function $M(r)$ or to Wiman--Valiron type concentration; a minimal counterexample to the affirmative liminf conjecture would be an entire function whose boundary modulus does not admit such concentration at infinitely many radii.